VITALI SEPETNITSKY

22/05 /2013

Research Current Status

Background

Classical WA* algorithm was takenDifferent reopening policies (currently, the

radical): Always Reopen (AR) No Reopen (NR)

It sounds reasonable that any solution found by the “AR” policy it at least “good”(*) (or even better) as any solution found by the “NR” policy

(*) Measured by cost of the found path and number of expanded states

Experiments

Korf’s 100 instances of 15-puzzle were takenKorf’s example weights were takenWA* with “AR” and “NR” policies was ran in

order to solve each instance (using the weights)

In the results we can see a lot of runs in which WA* with “NR” policy outperforms WA* with “AR” policy!

This contradicts our assumption!

More detailed analysis

By running the same test on: 15-puzzle 9-puzzle 3x2-puzzle

1. The phenomenon described above can appear with any instance – there are no specific instances

2. The phenomenon appears mostly in around 4-5

3. As the weight grows, the improvement of “NR” over “AR” grows too

A toy example

Strange!

Moreover, let’s look on this graph:

Sh=2

4

Bh=2

Ch=4

Dh=3

Eh=4

Gh=0

4

4

40 5

Kh=4

4

4

61

D3h=4

1

S1h=4

S2h=4

S3h=4

S4h=4

S5h=4

6

6

6

6

6

A toy example (1)

We will show 4 different cases by simply changing the weight of WA*

Found solution cost

Lower for AR

Lower for NR

# of expand

ed states

Lower for AR

Lower for NR

Sh=2

4

Bh=2

Ch=4

Dh=3

Eh=4

Gh=0

4

4

40 5

Kh=4

4

4

61

D3h=4

1

S1h=4

S2h=4

S3h=4

S4h=4

S5h=4

6

6

6

6

6

A toy example (2): Case 1

“NR” produces a better solution cost“NR” generates and expands LESS states

Solving using “AR” : Solving using “NR” : Path found : [S,C,D,G] Path found : [S,B,K,G]Path cost : 45 Path cost : 12Generated : 28 Generated : 25Expanded : 12 Expanded : 11

See Run

A toy example (3): Case 2

“NR” produces a better solution cost“NR” generates and expands MORE states

Solving using “AR” : Solving using “NR” : Path found : [S,C,D,G] Path found : [S,B,K,G]Path cost : 45 Path cost : 12Generated : 22 Generated : 25Expanded : 6 Expanded : 11

A toy example (4): Case 3

“AR” produces a better solution cost“AR” generates and expands LESS states

Solving using “AR” : Solving using “NR” : Path found : [S,C,D,G] Path found : [S,B,D,G]Path cost : 45 Path cost : 48Generated : 22 Generated : 23Expanded : 6 Expanded : 10

A toy example (5): Case 4

“AR” produces a better solution cost“AR” generates and expands MORE states

Solving using “AR” : Solving using “NR” : Path found : [S,C,D,G] Path found : [S,B,D,G]Path cost : 45 Path cost : 48Generated : 22 Generated : 18Expanded : 6 Expanded : 5

Some Results

9-puzzle15-puzzle

(2x3-puzzle yields the same results)

Distribution - the instances set

0 10 20 30 40 50 60 70 80 90 1000

2

4

6

8

10

12

14

NR better than AR in different instances

Instance Number

Nu

mb

er

of

inst

an

ces

wit

h

NR

bett

er

than

AR

9-puzzle 15-puzzle

0 10 20 30 40 50 60 70 80 90 1000

5

10

15

20

25

NR better than AR in different instances

Instance NumberNu

mb

er

of

inst

an

ces

wit

h

NR

bett

er

than

AR

Distribution - different weights

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

NR better than AR with different weights

Wh/WgNu

mb

er

of

inst

an

ces

wit

h N

R

Bett

er

than

AR

9-puzzle 15-puzzle

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

NR better than AR with different weights

Wh/Wg

Nu

mb

er

of

inst

an

ces

wit

h

NR

Bett

er

than

AR

Distribution – depth improvement

9-puzzle 15-puzzle

0 5 10 15 20 25 30 35 40 45 500

5

10

15

Average difference be-tween AR depth and NR

depth with different weights

wh/wg

Ave

rag

e d

iffere

nce

betw

een

A

R d

ep

th a

nd

NR

dep

th

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

Average difference be-tween AR depth and NR

depth with different weights

wh/wg

Ave

rag

e d

iffere

nce

betw

een

A

R d

ep

th a

nd

NR

dep

th

Distribution over 4-cases

Number of different runs(run = instance (#) + weight)

2x3-puzzle 9-puzzle 15-puzzle(Case 1)

NR-dep < AR-depNR-exp+gen < AR-exp+gen

28 97 413

(Case 2)

NR-dep < AR-depNR-exp+gen > AR-exp+gen

16 66 406

(Case 3)

NR-dep > AR-depNR-exp+gen < AR-exp+gen

104 187 568

(Case 4)

NR-dep > AR-depNR-exp+gen > AR-exp+gen

99 147 579

avg: 61.75sdev: 46.20

avg: 124.25sdev: 54.51

avg: 491.5sdev: 94.83